Averaging all losses above a VaR threshold

Identify the subset used

For the original 90% confidence model, VaR is 10. Strictly above it selects only the 5% outcome at 50; at or above it selects 30% probability at losses 10 and 50. Neither subset equals the desired worst 10% without a boundary adjustment.

Allocate the exact mass

Include all 5% at loss 50, then 5% of the available 25% at loss 10. The weighted sum is 3 and the denominator 0.10, giving ES 30. The boundary loss remains 10; only its included probability is fractional.

Audit the denominator

Avoid an unqualified universal shortcut

A no-atom continuous model can justify familiar conditional-tail expressions. That does not remove the need to state the convention when the input record is discrete.